Abstract. In this work we investigate the derivations of n−dimensional complex evolution algebras, depending on the rank of the appropriate matrices. For evolution algebra with non-singular matrices we prove that the space of derivations is zero. The spaces of derivations for evolution algebras with matrices of rank n − 1 are described.
Abstract. The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras the criterium of nilpotency is established in terms of the properties of corresponding matrices. Moreover, it is proved that for nilpotent n-dimensional complex evolution algebras the possible maximal nilpotency index is 1 + 2 n−1 .
The present paper is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz n-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz n-algebra and Cartan subalgebras and regular elements of the corresponding factor n-Lie algebra is established.
We study complex finite-dimensional Leibniz algebra bimodule over [Formula: see text] that as a Lie algebra module is split into a direct sum of two simple [Formula: see text]-modules. We prove that in this case there are only two nonsplit Leibniz [Formula: see text]-bimodules and we describe the actions.
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